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Connection formulae for the first Painlevé transcendent in the complex domain. (English) Zbl 0777.34007
The authors study the first Painlevé equation (1) $y''=6y\sp 2+x$ and the auxiliary linear ordinary differential equation $${d \psi \over d\lambda}= \left\{(4\lambda\sp 2+x+2y\sp 2) \sigma\sb 3-i(4y \lambda\sp 2+x+2y\sp 2)\sigma\sb 2- \left(2y'\lambda+{1 \over 2\lambda} \right) \sigma\sb 1\right\} \psi. \tag 2$$ They then go on using the isomonodrome deformation method to find a complete asymptotic descriptions of the first Painlevé transcendent in the complex domain by relating equations (1) and (2). The starting point is a result due to M. Jimbo and T. Mirva, whereby an analytic function $y(x)$ is a solution of equation (1) if and only if the monodromy data of equation (2) do not depend on $x$. The authors then explore this relationship using the Stokes multipliers $s\sb k$. A very elegant treatment is then presented whereby asymptotic descriptions are presented for the solutions of the equation (1) connecting to the Stokes multiplier constraints in sectors of the complex plane. Several theorems are proven such as if $y(x)$ is a solution of (1) subject to Stokes multiplier constraints $s\sb{5- 2\ell}=0$, then in the sector $\Omega\sb \ell:{3\pi\over 5}+{2\pi\over 5}\ell<\varphi=\text{arg} x<{7\pi\over 5}+{2\pi\over 5}\ell$, $\ell=0$, $\pm 1,\dots$, the solution $y(x)$ has asymptotics $$y(x)=\mu\sum\sp \infty\sb{n=0}a\sb nu\sp{-sn}+{\cal O}(u\sp{-\infty})\text{ where } u=\vert x/6\vert\sp{1/2}\exp \left[i \left({\varphi-\pi \over 2}-\pi \ell \right)\right]\text{ as } \vert x\vert\to\infty\text{ in } \Omega\sb \ell.$$

##### MSC:
 34M55 Painlevé and other special equations; classification, hierarchies 34E20 Asymptotic singular perturbations, turning point theory, WKB methods (ODE) 34E05 Asymptotic expansions (ODE) 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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##### References:
 [1] Garnier, R.,Ann. Sci. Ecole Norm. Sup. 29, 1-126 (1912). · Zbl 43.0382.01 [2] Jimbo, M. and Miwa, T.,Physica D 2, 407 (1981). · Zbl 1194.34166 · doi:10.1016/0167-2789(81)90021-X [3] Kapaev, A. A.,Differential Equations 24, 1107 (1988). [4] Joshi, N. and Kruskal, M. D., Connection results for the first Painlev? equation, School of Math. Univ. of New South Wales, Applied Math. Preprint, AM 91/4 (1991). [5] Kitaev, A. V., Mathematical problems of wave propagation theory. 19,Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov 179, 101-109 (1989). · Zbl 0708.34059 [6] Its, A. R. and Novokshonov, V. Yu.,The Isomonodromic Deformation Method in the Theory of Painlev? Equations, Lecture Notes in Math. 1191, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1986. [7] Boutroux, P.,Ann. Sci. Ecole Norm. Sup. 30, 265-375 (1913);31, 99-159 (1914). [8] Joshi, N. and Kruskal, M. D.,Phys. Lett. A 130, 129-137 (1988). · doi:10.1016/0375-9601(88)90415-X